Section 4.9
Laplace’s equation in cylindrical coordinates
As in the case of spherical coordinates, this equation is solved by a series expansion in terms of products of functions of the
individual cylindrical coordinates. That is, we use separation of variables.
Φ
(
ρ, ϕ, z
)=
X
l,m,n
α
lmn
U
l
(
ρ
)
V
m
(
ϕ
)
W
k
(
z
)
.
(4.53)
Substituting
Φ
lmk
(
ρ, ϕ, z
U
l
(
ρ
)
V
m
(
ϕ
)
W
k
(
z
)
.
into Eq 4.52 and dividing by
Φ
lmk
(
ρ, ϕ, z
)
, we can sequentially
separate all the variable .dependence:
1
U
l
ρ
d
dρ
·
ρ
d
dρ
U
l
¸
+
1
V
m
ρ
2
d
2
dϕ
2
V
m
=
−
1
W
n
d
2
dz
2
W
k
=
±
k
2
(4.54)
ρ
U
l
d
dρ
·
ρ
d
dρ
U
l
¸
∓
ρ
2
k
2
=
−
1
V
m
d
2
dϕ
2
V
m
=
±
m
2
(4.55)
These functions satisfy the Eqs. 4.54, 4.55 and 4.56:
d
2
W
k
(
z
)
dz
2
=
λ
z
W
k
(
z
)
;
λ
z
=
k
2
(4.54a)
W
k
(
z
A
k
e
kz
+
B
k
e
−
kz
(4.54b)
Note that
k
can be complex, pure imaginary or real.
d
2
dϕ
2
V
m
(
ϕ
λ
ϕ
V
m
(
ϕ
)
;
λ
ϕ
=
−
m
2
(4.55)
V
m
(
ϕ
C
m
e
imϕ
+
D
m
e
−
imϕ
(4.55b)
Note that
m
could be noninteger and/or complex
ρ
2
d
2
dρ
2
U
(
ρ
)+
ρ
d
dρ
U
(
ρ
¡
k
2
ρ
2
−
m
2
¢
U
(
ρ
)=0
(4.56)
(
kρ
)
2
d
2
d
(
)
2
U
m
(
d
d
(
)
U
m
(
¡
(
)
2
−
m
2
¢
U
m
(
U
m
(
a
m
J
m
(
b
m
N
m
(
)
(4.56b)
wherenowthelabelon
U
m
(
)
becomes
m
and the
J
m
and .
N
m
are called Bessel functions of the
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This note was uploaded on 12/19/2010 for the course PHYS 411 taught by Professor G during the Spring '10 term at Missouri S&T.
 Spring '10
 G

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